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arXiv:1807.02858 · Annals of Physics

Exact rotating wave approximation

Daniel Zeuch, Fabian Hassler, Jesse J. Slim, David Peter Divincenzo

Published Oct 26, 2020• 63 citations
arXivHTMLPDF
DOI
10.1016/j.aop.2020.168327

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The Jaynes-Cummings model without the rotating-wave approximation can be solved exactly by an extended Swain ansatz with conserved parity. Analytical approximations are then performed at different levels. The well-known rotating-wave approximation (RWA) is naturally covered in the present zeroth- and first-order approximations. A first-order correction to the RWA can be obtained in a second-order approximation, by which the effect of the counter-rotating-wave term emerges clearly. Concise analytical expressions are given explicitly and can be applicable up to the ultrastrong-coupling regime. A preliminary application to vacuum Rabi splitting is shown to be very successful.
The system of one harmonic oscillator interacting with a continuum is studied. The problem is solved for two cases: the exact (full) Hamiltonian and the truncated Hamiltonian [the rotating-wave approximation (RWA)]. Comparing these two solutions, we formulate the conditions of the applicability of the RWA. The case in which the solution can be treated as Markovian is also discussed. The conditions for the existence of discrete levels are obtained. It was shown that under certain conditions, nonoscillating terms in the solution, leading to the instability of the whole system, can exist. Such terms cannot be obtained within either the Markovian or the rotating-wave approximation. Two particular cases of coupling are proposed and the results of numerical simulations are discussed.
The generalized rotating-wave approximation (GRWA) is presented for the two-qubit quantum Rabi model. The analytical expressions in the zeroth-order approximation recover the previous adiabatic ones. The counter-rotating-wave terms can be eliminated by performing the first-order corrections. An effective solvable Hamiltonian with the same form as the ordinary RWA one is then obtained, giving very accurate eigenvalues and eigenstates. Energy levels in the present GRWA are in accordance with the numerical exact diagonalization ones in a wide range of coupling strengths. The population dynamics in the GRWA are in quantitative agreement with the numerical results and exhibit the absence of collapses clearly, revealing the effects of the counter-rotating wave.